Coexistence lines Clapeyron equations

The integration path is largely set by the definition of the problem one knows beforehand the plane in which the coexistence line is desired. However application of simple transformations to the field variables can be beneficial if they reshape the coexistence line into a simpler form. The advantages can be improved accuracy, precision, and stability of the integration. A very familiar example is the conversion from pressure P to ln(P) in the characterization of vapor-liquid coexistence. The Clapeyron equation in the latter instance is... 

Once a point on the coexistence line has been found, one can trace out more of it using the approach of Kofke [177. 178] to numerically integrate die Clapeyron equation... 

Vaporization Transition Clausius-Clapeyron Equation For the liquid-vapor coexistence line ( vapor-pressure curve ), the Clapeyron equation (7.29) becomes... 

In accordance with the Clapeyron equation and Le Chatelier s principle, the more highly ordered (low-entropy) phases tend to lie further to the left (at lower 7), whereas the higher-density phases tend to lie further upward (at higher 7). The mnemonic (7.32) allows us to anticipate the relative densities of adjacent phases. From the slope, for example, of the ice II-ice III coexistence line (which tilts forward to cover ice III), we can expect that ice II is denser than ice III (pn > pm). Similarly, from the forward slopes of the liquid coexistence lines with the high-pressure ices II, V, and VI, we can expect that cubes of ice II, ice V, and ice VI would all sink in a glass of water, whereas ice I floats (in accord with the backward tilt of its phase boundary). Many such inferences can be drawn from the slopes of the various phase boundaries in Fig. 7.3, all consistent with the measured phase densities Pphase (in gL 1), namely,... 

The first of these can be recognized as the ordinary Clapeyron equation for a pure two-phase system (usually written for equimolar phases Af(1) = Af(2) = 1 cf. Sections 7.2.2 and 11.11), and the second is an analogous equation determining the slope of the coexistence curve in the pi-T plane. These equations in turn determine the slope of the coexistence line in the pi-P plane ... 

Once a state point of coexistence is established, additional state points can be determined expeditiously through application of the Gibbs-Duhem integration method [48,85,86]. In this approach a differential equation for the coexistence line is used to guide the establishment of state points away from the known coexistence point. The most well known such formula is the Clapeyron equation [41]... 

After calculating the AH and AV for PEO/P(EO-b-DMS) system, we know that just as the Clausius-Clapeyron equation(Ma, 1982) predicted the coexistence line of two phases in a T-P plane of PEO/P(EO-b-DMS) system according to (dT / dP = TAI4, / Since AH, ... 

Notice that this derivative is along the coexistence line in the T-P-plane. Therefore we can apply the Clapeyron equation, i.e. 

Barrer s discussion4 of his analog of Eq. 28 merits some comment. Equation 28 expresses the equilibrium condition between ice and hydrate. As such it is valid for all equilibria in which the two phases coexist and not only for univariant equilibria corresponding with a P—7" line in the phase diagram. (It holds, for instance, in the entire ice-hydratell-gas region of the ternary system water-methane-propane considered in Section III.C.(2).) In addition to Eq. 28 one has Clapeyron s equation... 

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